Group renaming in iMessage is fundamentally broken. If I rename a group of people on my device, that group gets renamed on all participants device. That's a problem for a couple of reasons. First, the relationships of participants isn't the same for everyone. For example, a message thread of myself and my parents might be labeled "Parents" on my device, and "Family" or "Son and Wife" on my father's. Second, the name of the group may be sensitive. If I'm on a thread with people I don't particularly like, maybe I rename it to "Idiots". Under the current implementation all users would see that.